PREVIEW OF CHAPTER I
Solving Quadratic Equations by
Iteration
A highlight of the introductory algebra curriculum is the
solution of the quadratic equation
ax2 + bx +
c = 0
by means of the
quadratic formula
 . |
| In fact, however, the
quadratic formula may be somewhat overrated. For in "solving"
the
equation |
 |
in the form |
 |
|
| we do not obtain a numerical
answer of the kind that enables us to locate x on a number line. Instead,
we obtain a
"solution in terms of radicals," one that requires evaluation of
a square root. But the problem of finding
square roots also calls for the solution of quadratic
equations, in this case |
x2 - 5 = 0.
So rather than
"solving" the original equation,
the quadratic formula merely reduces the problem to a simpler one -
i.e., that of solving a quadratic equation of the form
 . |
In this sample
lesson we will:
- Recount "the Babylonian method" for finding
the square root of k.
- Formulate this historical technique as an iterative process
of the form
|
 . |
- Give this iterative process a geometric
interpretation
in terms of "staircase diagrams."
- Show that such an iterative process can be used to
solve quadratic equations of the form ax2 + bx + c = 0
directly - i.e., without calling on the litany:
minus bee, plus
or minus the square root of bee squared minus four ay cee, all over two
ay. |
| |
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